TU Kaiserslautern - Fachbereich Mathematik AG Algebra, Geometrie und Computeralgebra Prof. Dr. W. Decker, Prof. Dr. M. Schulze, Dr. Janko Böhm

Tel.: +49 (0)631/205-2730 Zi. 435, 434, 430, Geb. 48 D-67663 Kaiserslautern

decker@mathematik.uni-kl.de mschulze@mathematik.uni-kl.de boehm@mathematik.uni-kl.de

Seminar: Algorithms in toric and algebraic geometry

Summer Term 2013

News

  The seminar will be Thursdays, 15:30-17:00 in 48-538. We will begin on 02.05.13.

  We will have an organizational meeting on Friday 19.04.13 at 13:45 - 15:15 in 48-538.

  Please register online in the URM system.

Goals

The seminar will provide a systematic introduction to toric geometry and algorithms in toric and algebraic geometry. Toric varieties are special type of algebraic varieties which are particularly easy to handle. The reason is that they are described by convex polyhedral objects, from which lots of interesting information can be obtained in a direct manner (in contrast to an arbitrary algebraic variety where it is very hard to compute). As an example, the affine 2-dimensional space ² can be represented as a toric variety by the positive orthant with the standard integer lattice:

Specifically, we will take a look at the following topics:

 Convex polyhedral cones, affine toric varieties (Adrian Koch)
 Projective toric varieties, fans (Peter Chini)
 Divisors (Raul Epure)
 Homogeneous coordinate rings (Tien Mai Nguyen and Bin Nguyen)
 Resolution of singularities
 Cohomology of line bundles (Sasha Valentinova Bagryanova)
 Betti numbers of toric varieties (Corina Birghila)
 Toric intersection theory (Hiep Dang).
 Riemann-Roch theorem (Florian Diebold).
 Bezout's theorem (Sebastian Muskalla).
 Stanley's theorem.
 Algebraic analogues of toric concepts.
 Birational geometry.

Each participant will give a talk in the seminar. There are topics which are more theoretical or more practical in nature. The practical ones come with some implementation work in Singular, which then will be demonstrated in the talk.

The targeted audience are students who have attended the lecture Computer Algebra or the lecture Commutative Algebra in the winter term, or who are attending the lecture Algebraic Geometry in the current summer term.

However, anyone with a basic knowledge of commutative algebra is welcome. Knowledge in algebraic geometry is not required.

The seminar is a good starting point if you plan to write any sort of thesis in computer algebra or algebraic geometry.

Reference books

Work load

Module description for the Bachelor courses in Mathematics.

 2 SWS seminar
 3 credit points

 30h contact hours for the seminar
 60h self-study hours

Seminar

The seminar will be Thursdays, 15:30 - 17:00 in 48-538.

We will begin on 02.05.13.

Please register online in the URM system. Write an email to boehm@mathematik.uni-kl.de if you have any issues with the date.
There will be organizational meeting in the first week of the semester.

Grading

Anyone is wellcome to listen to the seminar. To obtain credit points you will have to give a talk.

Computer Algebra

Singular:

The open source computer algebra system Singular is being developed in Kaiserslautern. It is one of the leading systems for calculations in polynomial rings. Algorithms implemented in Singular deal with Gröbner bases, free resolutions, polynomial factorization, primary decomposition, and many other problems in commutative algebra.

You can download Singular here, read the online manual is here, and obtain the source code here.

Polymake:

The open source computer algebra system Polymake is one of the leading systems for calculations in convex geometry. It has facilities for visualization and can be accessed directly from Singular.

You can download Polymake here, read the online manual is here.

Contact information